Too Big to Fail: a Quick Cost/Benefit Sketch (Super Nerdy) Part I

One of the most important things we have to address when it is all said and done is that how do we deal with companies that are "too big to fail".

Let's first define formally what too big to fail means:

SystemCost(Inc(z,fail)|no bailout, x) - SystemCost(Inc(z,fail)|bailout, x) > epsilon(z,x)

In English: given that a company of size z fails, the cost of offering no bailout to the system is significantly higher than the cost of offering bailout. So Enron was not "too big to fail" but Lehman almost was, even though the amount of money involved in both companies are beyond the imagination of normal human beings.

Now let's call epsilon(z, x), then, as systemic cost potential. Essentially, companies that has a very large epsilon(z, x) can take an economy hostage. x denotes all other variables that parametrize epsilon. For example, let's accept the supposition that a company like McDonald's have an epsilon of 0 (zero), regardless of what z is.

That is to say, even if it controls 90% of the world's restaurant business, a sudden bankruptcy will not unduly destroy the economy as the remaining 10% immediately meet the restaurant demand and no one else goes bankrupt. Note that this example is probably not entirely true since some of McDonald's suppliers would go under as well, but I'm just using this example as a matter of illumination.

So given the this systemic cost potential, what should the "optimal size" be?

Using a very straightforward cost-benefit analysis, that boils down to:

NetBenefit=Benefit(z,x) -Cost(z,x) - E[epsilon(z, x)]

From the systemic view. The "optimal" size z from the perspective of the economy is such that NetBenefit is maximized with respect to size z. Let's call this optimal z as z*

The problem as it stands, of course, is that bank's own cost/benefit analysis is:

NetBenefit=Benefit(z,x) -Cost(z,x)

And a bank will therefore chose a level z** that will be bigger than z*. Inefficiently so.

In English, the phrase "privatize the gains, but socialize the losses" basically describes what is happening.

One takeaway is that, z**, which I will call it "disastrous capitalist decision" for expository clarity (however polemical that may be), will happen in an unregulated free market, because this is a case of cost externality market failure.

So what is the optimal z*? Without going too much into it, calculating that is at this point is either ridiculously difficult to Black Swan impossible. I think I will post a followup on this once I can wrap my head around it.